Graded contractions on the orthogonal Lie algebras of dimensions 7 and 8

TL;DR

This paper classifies graded contractions of specific $ ext{so}(7)$ and $ ext{so}(8)$ Lie algebras with $ ext{Z}_2^3$-gradings, revealing new solvable Lie algebras and generalizing previous classifications on $ ext{g}_2$.

Contribution

It provides a comprehensive classification of graded contractions on $ ext{so}(7)$ and $ ext{so}(8)$ with $ ext{Z}_2^3$-gradings, extending earlier work on $ ext{g}_2$.

Findings

Discovery of two large families of solvable Lie algebras of dimensions 21 and 28.

Classification results up to two notions of equivalence.

Generalization potential to any good $ ext{Z}_2^3$-grading on arbitrary Lie algebras.

Abstract

Graded contractions of certain non-toral $\mathbb{Z}_2^3$-gradings on the simple Lie algebras $\mathfrak{so}(7,\mathbb C)$ and $\f{so}(8,\mathbb C)$ are classified up to two notions of equivalence. In particular, there arise two large families of Lie algebras (the majority of which are solvable) of dimensions 21 and 28. This is achieved as a significant generalization of the classification of related graded contractions on $\mathfrak g_2$, the derivation algebra of the octonion algebra. Many of the results can be further extended to any \emph{good} $\mathbb Z_2^3$-grading on an arbitrary Lie algebra.